A 9-Minute Technique Intended for PTC-209

(12)Theorem one �� MIMO affine nonlinear procedure is A 1-Minute Cheat For PTC-209described asx�B=f(x)+��i=1mgi(x)ui,y=h(x).(13)Set the equilibrium point x0 X. The problem of MIMO nonlinear process feedback linearization can have an answer, if and only if a neighborhood U near x0 and m functions h6(x),��, hm(x) exists. The remedy can make relative degree (r1,��, rm) of nonlinear program The 15-Sec Strategy For the PTC-209 (eleven) satisfy r1 + r2 + +rm = n, in which n denotes the dimension of process and m stands for input dimension.2.three. Demonstration of Definition 1Define the equilibrium point of maglev procedure x0=[z10,z�B10,z20,z�B20,i10,i20]T, and suppose state variables x1, x2, x5, x6 R+, according to serious maglev program construction.

Compute the vector field as follows, by f(x) and g(x):Lg1Lf0h6(x)=0,??Lg2Lf0h6(x)=0,Lg1Lfh6(x)=0,??Lg2Lfh6(x)=0,Lg1Lf2h6(x)=2A��x5��0N��0,??Lg2Lf2h6(x)=2B��x6��0N��0,Lg1Lf0h6(x)=0,??Lg2Lf0h6(x)=0,Lg1Lfh6(x)=0,??Lg2Lfh6(x)=0,Lg1Lf2h6(x)=2B��x5��0N��0,??Lg2Lf2h6(x)=2A��x6��0N��0.(14)Namely, above equations satisfyLgjLfkhi(x)=0,????x��V,??1��j��m,??1��i��m,0��k��ri?2,LgjLfrihi(x)��0,(15)whereH(x)=[Lg1Lfr1?1h6(x)Lg2Lfr1?1h6(x)Lg1Lfr2?1h6(x)Lg2Lfr2?1h6(x)]=[2A��x5��0N2B��x6��0N2B��x5��0N2A��x6��0N].(sixteen)With the moment, det H(x) = 4(A2 ? B2) �� x5x6/��02N2 �� 0; then E(x) isn't singular. Relative degree of maglev program isThe 5-Sec Strategy For PTC-209 (r1, r2) = (three,three), andr=��i=12ri=6=n.(17)As a result, Theorem 1 has been proved.2.four. MIMO Double Suspension Factors Model with Suggestions LinearizationCompute the vector field generated by f(x) and h(x) as follows:Lf3h6(x)=1��02N2[?AR(3x1+x3)x52?BR(x1+3x3)x62],Lf3h6(x)=1��02N2[?BR(3x1+x3)x52?AR(x1+3x3)x62].

(18)Design feedback control worth u = H?one(x)[?b(x) + v], where the terms u = [u1u2]T, v = [v1v2]T, andH?1(x)=��0N2(A2?B2)��[Ax5?Bx6?Bx5Ax6],b(x)=[Lfr1h6(x)?Lfr2h6(x)]T=[Lf3h6(x)?Lf3h6(x)]T.(19)We will receive the control value after linearization from (19):u=H?one(x)[?b(x)+v]={��0N2(A2?B2)��[?Ax5��Lf3h6(x)+Bx6��Lf3h6(x)+Ax5��v1?Bx6��v2]��0N2(A2?B2)��[Bx5��Lf3h6(x)?Ax6��Lf3h6(x)?Bx5��v1+Ax6��v2].(20)The diffeomorphic mapping matrix is��(x)=[h6(x)Lfh6(x)Lf2h6(x)h6(x)Lfh6(x)Lf2h6(x)]=[x1x2A��x52S��0+B��x62S��0+C��f1+D��f2+gx3x4B��x52S��0+A��x62S��0+D��f1+C��f2+g].(21)Choose the coordinates of transformation by matrix ��(x):��=[h6(x)Lfh6(x)Lf2h6(x)h6(x)Lfh6(x)Lf2h6(x)]T.(22)In sum, maglev control system model after linearization is showed as follows:�ĨB=����+Tv,y=����,(23)where��=[010000001000000000000010000001000000],??T=[000010000001],��=[100000000100].(24)Maglev system after feedback linearization is expressed by two-level integral subsystems:G(s)=[1s3001s3].(25)Double suspension points' closed control block diagram with magnetic flux feedback is showed in Figure 3.Figure 3Double suspension points' control construction diagram with feedback linearization.3.